Structure of Dark Matter Halos in Warm Dark Matter models and in models with Long-Lived Charged Massive Particles

0.98 MB·English

by  Ayuki Kamada

IPMU13-0012
Structure of Dark Matter Halos in Warm Dark Matter models and in models
with Long-Lived Charged Massive Particles
3
Ayuki Kamada, Naoki Yoshida
1
0
2 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa,

n Chiba 277-8583, Japan
a
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
J

3
1 [email protected]

] Kazunori Kohri
O
C
Cosmology group, Theory Center, IPNS, KEK, and The Graduate University for Advanced Study
.
h (Sokendai), Tsukuba, 305-0801, Japan
p
-
o and
r
t
s Tomo Takahashi
a
[

Department of Physics, Saga University, Saga 840-8502, Japan
1
v
4
ABSTRACT
4
7
2
. Westudytheformationofnon-linearstructuresinWarmDarkMatter(WDM)mod-
1
0 els and in a Long-Lived Charged Massive Particle (CHAMP) model. CHAMPs with a
3
decay lifetime of about 1yr induce characteristic suppression in the matter power spec-
1
: trum at subgalactic scales through acoustic oscillations in the thermal background. We
v
i explore structure formation in such a model. We also study three WDM models, where
X
the dark matter particles are produced through the following mechanisms: i) WDM
r
a particles are produced in the thermal background and then kinematically decoupled;
ii) WDM particles are fermions produced by the decay of thermal heavy bosons; and
iii) WDM particles are produced by the decay of non-relativistic heavy particles. We
show that the linear matter power spectra for the three models are all characterised
by the comoving Jeans scale at the matter-radiation equality. Furthermore, we can
also describe the linear matter power spectrum for the Long-Lived CHAMP model in
terms of a suitably defined characteristic cut-off scale k , similarly to the WDM mod-
Ch
els. We perform large cosmological N-body simulations to study the non-linear growth
of structures in these four models. We compare the halo mass functions, the subhalo
mass functions, and the radial distributions of subhalos in simulated Milky Way-size
halos. For the characteristic cut-off scale k = 51hMpc−1, the subhalo abundance
cut
– 2 –
(∼ 109M ) is suppressed by a factor of ∼ 10 compared with the standard ΛCDM
sun
model. We then study the models with k ≃ 51, 410, 820hMpc−1, and confirm that
cut
thehalo andthesubhaloabundancesandtheradialdistributions of subhalosareindeed
similar between the different WDM models and the Long-Lived CHAMP model. The
result suggests that the cut-off scale k not only characterises the linear power spectra
cut
butalso can beusedto predictthenon-linearclustering properties. Theradial distribu-
tion of subhalos in Milky Way-size halos is consistent with the observed distribution for
k ∼ 50−800hMpc−1; such models resolve the so-called “missing satellite problem”.
cut
Subject headings: cosmology: theory - early universe - dark matter
1. Introduction
Theprecisemeasurementof thecosmic microwave background (CMB)anisotropies established
the standard Λ + Cold Dark Matter (ΛCDM) cosmology (Komatsu et al. 2011). Observations of
the large-scale structure of the Universe, such as the galaxy power spectra from the Sloan Digital
SkySurvey(SDSS)alsoconfirmeditssuccessinpredictingthelargescalestructuresoftheUniverse
(e.g. Tegmark et al. (2004); Reid et al. (2010); Percival et al. (2010)).
The validity of the ΛCDM model on the galactic and the subgalactic scales has long been
caught up in debate. Moore et al. (1999) argue that the number of dark matter subhalos is
10 − 100 times larger than the number of satellites observed around the Milky Way(Kravtsov
2010). The so-called “missing satellite problem” has been revisited in a somewhat quantitative
context (Boylan-Kolchin et al. 2011; Lovell et al. 2012; Boylan-Kolchin et al. 2012). For exam-
ple, Boylan-Kolchin et al. (2011) argue that, in the ΛCDM model, ∼ 10 most massive subhalos
in a galactic halo are too concentrated to be consistent with the kinematic data for the bright
Milky Way satellites. Also, observations of the rotation velocities of galaxies using the 21cm line
by Papastergis et al. (2011) show that the abundance of galaxies with observed velocity width
w = 50kms−1 is ∼ 8 times lower than predicted in the ΛCDM model.
ItisoftensuggestedthatWDMmodelsresolvetheapparentproblemsonsubgalacticscales(Bode et al.
2001). WDM particles have non-negligible velocity dispersions,which act as an effective “pressure”
of the WDM fluid. Essentially, the subgalactic-scale density fluctuations are suppressed. The re-
sultant matter power spectrum is quickly reduced around the cut-off scale that is determined by
the velocity dispersion. Motivated by the recent interest in this problem, several authors study
the structure formation in WDM models(Dunstan et al. 2011; Schneider et al. 2012; Menci et al.
2012).
ConstraintsonWDMmodelscanbeobtained fromastronomical observations. Observationsof
Lyman-α forests are often used for the purpose(Viel et al. 2005; Boyarsky et al. 2009). Absorption
features in quasar spectra reflect the number density of neutral hydrogen, from which we can esti-
– 3 –
matethematterpowerspectrumalongthelineofsight, evenatlargewavenumbersk ∼ 10hMpc−1.
WDM models have also interesting implications for the cosmic reionization (Barkana et al. 2001;
Yoshida et al.2003;Gao & Theuns2007). Theformationofthefirstobjects,andhencetheproduc-
tion ofionizing photons,aredelayed inWDMmodels. Ontheother hand,WDM modelscouldhelp
the completion of the cosmic reionization. Yue & Chen (2012) suggest that the reduced number
of subhalos in WDM models makes the recombination of ionized hydrogens inefficient and results
in earlier completion of the cosmic reionization. It is clearly important to study the clustering
properties in WDM models in both linear and non-linear evolution regimes.
There are also renewed interest in particle physics. Several candidates for WDM are suggested
in particle physics models beyond the Standard Model, such as light gravitinos(Kawasaki et al.
1997),sterileneutrinos(seeKusenko(2009)forareviewandreferences)andsuperWIMPs(Cembranos et al.
2005). It is important to notice that WDM particles can be produced via different mechanisms.
Nevertheless, theaboveconstraints fromastronomicalobservations arefocusedonasinglequantity,
e.g., the mass of WDM particle in a specific model. It is unclear if such constraints can be applied
to WDM models with different production mechanisms. Detailed comparisons of a wide class of
models are clearly needed.
In this paper, we also consider a Long-Lived CHAMP model. Throughout this paper, we
assume that CHAMPs have an elementary charge, either positive or negative. CHAMPs are gen-
erally realized in models beyond the Standard Model of particle physics. One such example is
a slepton, a superpartner of leptons in supersymmetric models. Sleptons as the lightest super-
symmetry particles (LSPs) are stable when R-parity is conserved. The abundance of such stable
CHAMPs, however, is severely constrained by the searches in deep sea water(see Beringer et al.
(2012) for a review and references). CHAMPs can also be unstable; a CHAMP decay into neu-
tral dark matter and other decay products including at least one charged particle. For exam-
ple, the stau can be the next lightest supersymmetric particle (NLSP) when the gravitino is
the LSP(Buchmu¨ller et al. 2006). It is well-known that CHAMPs could affect the big bang nu-
cleosynthesis (BBN) reaction rates and thus change the abundance of light elements(Pospelov
2007; Kohri & Takayama 2007; Kaplinghat & Rajaraman 2006; Cyburt et al. 2006; Steffen 2007;
Hamaguchi et al. 2007; Kawasaki et al. 2007; Jedamzik 2008a,b; Jittoh et al. 2011). Several au-
thors(Sigurdson & Kamionkowski2004;Kohri & Takahashi2010)suggestthepossibilitythatCHAMPs
with a lifetime about 1yr can act effectively as WDM through acoustic oscillations in the thermal
background. We study the effect of the oscillations on the matter power spectrum.
We calculate the linear evolution of thematter density fluctuations for the three WDM models
andtheLong-LivedCHAMPmodel. WeshowthatthecomovingJeansscaleatthematter-radiation
equality characterises the linear matter power spectra in the three WDM models well. We use the
obtained linear matter power spectra as initial conditions of N-body simulations to follow the
non-linear evolution of the matter distribution. We compare the halo mass functions, the subhalo
mass functions, and the radial distributions of subhalos in Milky Way-size halos to discuss the
clustering properties in the WDM models and in the Long-Lived CHAMP model. We show that
– 4 –
these statistics are similar when the cut-off scale is kept the same. We find that the WDM models
and the Long-Lived CHAMP model with the characteristic cut-off scale k ∼ 50−800hMpc−1
cut
resolve the so-called “missing satellite problem”.
The rest of this paper is organised as follows. In Sec.2, we summarize three WDM models and
a Long-Lived CHAMP model we consider. Then, we introduce the common cut-off scale k which
cut
characterises the linear matter power spectra in these models. In Sec.3, after describing the details
of N-body simulations, we show simulation results and discuss their implications. Specifically, we
mention the similarity of these models with the same cut-off and the possibility that CHAMPs
behave like WDMs and resolve the “missing satellite problem”. Finally, in Sec.4, concluding
remarks are given.
Throughout this paper, we take the cosmological parameters that are given in Komatsu et al.
(2011) as the WMAP+BAO+H Mean; 100Ω h2 = 2.255, Ω h2 = 0.1126, Ω = 0.725, n =
0 b CDM Λ s
0.968, τ = 0.088 and ∆2(k ) =2.430×10−9, while we replace the energy density of CDM Ω h2
R 0 CDM
by the energy density of WDM Ω h2 for the WDM models and by the energy density of neutral
WDM
dark matter produced by the CHAMP decay for the Long-Lived CHAMP model.
2. WDM models and Long-Lived CHAMP model
In this section, we summarize three WDM models and a Long-Lived CHAMP model we con-
sider in this paper. We describe production mechanisms of WDM particles in each model and show
the exact shapes of the velocity distribution. In the following subsections, we focus on three WDM
models to specify our discussion, although our results can beapplied to any WDM models with the
same shapeof the velocity distribution. Then, we introducethe Jeans scale at the matter-radiation
equality. The matter power spectra in the three WDM models with the same Jeans scale at the
matter-radiation equality are very similar. Their initial velocity distributions affect the damping
tailofthematterpowerspectra. Wealsodescribetheevolution ofthelinearmatterdensityfluctua-
tions inaLong-Lived CHAMP model. Thematter power spectrumis truncatedaroundthehorizon
scale at thetime whenCHAMPsdecay. Interestingly, theresultingpower spectrumappearssimilar
to those in WDM models.
2.1. Thermal WDM
In this type of models, fermionic WDM particles are produced in the thermal background.
They are decoupled from the thermal background as the Universe expands and cools. At the
time of the decoupling, their momentum obeys the thermal distribution, that is, the Fermi-Dirac
distribution. We consider the generalized Fermi-Dirac distribution,
β
f(p)= . (1)
ep/TWDM +1
– 5 –
Here and in the following, p denotes the comoving momentum of WDM particles, and T is the
WDM
effective temperaturethat characterises the comoving momentum of WDM particles. In the case of
the light gravitino(Dine et al. 1996) and the thermally produced sterile neutrino(Olive & Turner
1982), T relates to the temperature of the left-handed neutrino T through the conservation
WDM ν
1/3
43/4
of the entropy, T = T where g is the effective number of the massless degrees
WDM (cid:16)gdec(cid:17) ν dec
of freedom at the decoupling from the thermal background. Note that β determines the overall
normalization of the momentum distribution and β = 1 in the case of the gravitino and the
thermally produced sterile neutrino. Dodelson & Widrow (1994) propose the sterile neutrino dark
matter produced via active-sterile neutrino oscillations. In this case, the active neutrinos in the
thermal background turn into the sterile neutrino via the coherent forward scattering(Cline 1992).
Theresultantmomentumdistributionofthesterileneutrinoisgivenbythegeneralized Fermi-Dirac
distribution(seeEq.(1))withT ≃ T andβ ∝ θ2M whereθ istheactive-sterile mixingangle
WDM ν m m
and M is the mass of the sterile neutrino.
2.2. WDM produced by the thermal boson decay
There are models in which the Majorana mass of the sterile neutrino arises from the Yukawa
couplingY withasingletboson(Shaposhnikov & Tkachev2006;Petraki & Kusenko2008). Inthese
models, the singlet boson couples to the Standard Model Higgs boson through an extension of the
Standard Models Higgs sector. The singlet Higgs boson has a vacuum expectation value (VEV) of
the order of the electroweak scale when the electroweak symmetry breaks down. When the sterile
neutrino is assumed to be WDM with a mass of an order of keV, the Yukawa coupling should
be very small Y ∼ O(10−8). This small Yukawa coupling makes the singlet boson decay to the
two sterile neutrinos when the singlet boson is relativistic and is in equilibrium with the thermal
background. Here, it should be noted that the sterile neutrino model is one specific example.
In WDM models, where relativistic bosonic particles in equilibrium with the thermal background
decay into fermionic WDM particles through the Yukawa interaction, WDM particles have the
same resultant momentum distribution (see Eq.(2) below). The resultant momentum distribution
is obtained by solving the Boltzmann equation(Boyanovsky 2008),
β
f(p)= g (p/T ) (2)
(p/T )1/2 5/2 WDM
WDM
where
∞ e−nx
g (x) = . (3)
ν nν
nX=1
Here, we have ignored the low momentum cut-off that ensures the Pauli blocking, while it does not
1/3
43/4
change our results. The effective temperature is given by T = T with the effective
WDM (cid:16)gpro(cid:17) ν
number of massless degrees of freedom at the production of the sterile neutrino g ∼ 100. The
pro
– 6 –
normalization factor β is determined by the Yukawa coupling Y and the mass of the singlet Higgs
boson M, β ∝ Y2M−1. The velocity distribution have an enhancement f ∝ p−1/2 at the low
B
momentum p/T ≪ 1, since the sterile neutrinos with lower momenta are produced by the less
WDM
boosted singlet boson, the decay rate of which is larger due to the absence of the time dilation.
This enhancement indicates the “colder” (than the thermal WDM) property of the sterile neutrino
dark matter produced by the decay of the singlet heavy boson.
2.3. WDM produced by the non-relativistic particle decay
In this type of models, a non-relativistic heavy particle decays into two particles, one or both
of which become WDM. Supersymmetric theories realize this type of scenarios e.g. when the
LSP is the gravitino and the NLSP is a neutralino. The relic abundance of the NLSP neutralino is
determinedatthetimeofchemicaldecouplingbythestandardargument(Gondolo & Gelmini1991;
Griest & Seckel 1991). Eventually, the non-relativistic neutralinos decay into LSP gravitinos that
becomeWDM.Theparticlesproducedbythedecayofthemodulifieldsandoftheinflatonfieldsare
another candidates of this type of WDM(Lin et al. 2001; Hisano et al. 2001; Kawasaki et al. 2006;
Endo et al. 2006; Takahashi 2008). When we assume the heavy particle decays in the radiation
dominated era, the momentum distribution of the decay products is given by(Kaplinghat 2005;
Strigari et al. 2007; Aoyama et al. 2011),
β
f(p)= exp(−p2/T2 ), (4)
(p/T ) WDM
WDM
where T is given by T = P a(t )/a(t ) with the physical canter-of-mass momentum
WDM WDM cm d 0
P , the scale factor a(t) at the decay time t and at the present time t . We have defined t as
cm d 0 d
H(t = t )= 1/2τ where H(t) is the Hubble parameter and τ is the lifetime of the heavy particle.
d
2.4. Jeans scale at the matter-radiation equality
Now, we introduce two quantities to characterise the property of WDM. One is the present
energy density of WDM, Ω ≡ ρWDM| . Throughout this paper, we assume WDM particles
WDM ρcrit t=t0
account for all of the dark matter, letting Ω h2 = 0.1126. Another important physical scale is
WDM
the comoving Jeans scale at the matter radiation equality t ,
eq
4πGρ
M
k = a (5)
J r σ2 (cid:12)
(cid:12)t=teq
(cid:12)
(cid:12)
with the gravitational constant G. Here, ρ is the matter density and σ2 is the mean square of
M
the velocity of the dark matter particles (see Eq.(8) below). Dark matter particles with k ∼
J
100−1000Mpc−1 are usually called WDM and expected to resolve the “missing satellite problem”.
– 7 –
We note that, in the present paper, we do not consider whether or not a particular set of
Ω and k is in a viable region of the respective model. One such example is the gravitino
WDM J
WDM, a representative of the Thermal WDM model (see subsection2.1). This model has only two
parameters, the effective number of the massless degrees of freedom at the decoupling g and the
dec
gravitino mass m , to set Ω and k . When we assume k ≃ 30Mpc−1, these two parameters
3/2 WDM J J
are determined as g ≃ 1000 and m ≃ 1keV. The effective number of the massless degrees of
dec 3/2
freedom at the decoupling of the gravitino is at most g ∼ 200 in the Minimal Supersymmetric
dec
Standard Model (MSSM), and hence, another mechanism such as entropy production is needed to
explain the gravitino WDM(Ibe et al. 2011; Ibe & Sato 2012).
2.5. Linear matter power spectra and Normalized velocity distribution
We follow the evolution of the primordial adiabatic fluctuations for the three WDM models by
modifying suitably the public software, CAMB(Lewis et al. 2000). We adopt the covariant multipole
perturbationapproachforthemassiveneutrino(Ma & Bertschinger1995;Lewis & Challinor2002).
We replace the Fermi-Dirac distribution of the massive neutrino by the momentum distributions of
theWDMmodelsdiscussedabove. OurapproachisvalidwhentheWDMparticlesarekinematically
decoupled at the cosmic time of interest. The Jeans scale of interest is around k ∼ O(100)Mpc−1.
J
TheprimordialfluctuationofthiswavenumberentersthehorizonatT ∼ O(10)keV. Inalargeclass
of WDM models, WDM particles are kinematically decoupled before the QCD phase transition,
T ∼ 100MeV, and thus our calculation is valid. 1
QCD
For comparison, we calculate the normalized velocity distributions for the WDM models and
the linear matter power spectra extrapolated to the present time z = 0. The results are shown in
Fig.1 for k = 51Mpc−1, which correspond to m ≃ 2keV for the thermally-produced gravitino
J 3/2
WDM. Here, we have defined the dimensionless matter power spectra as,
1
∆(k) ≡ k3P(k) (6)
2π2
with the matter power spectra P(k).
The velocity distribution g(v) is normalized as follows:
∞
dvg(v) =1, (7)
Z
0
1TherearevariantsofWDMmodelsinwhichWDMparticlesareproducedbythenon-relativisticparticledecayat
late epochs. Thematter power spectrum could beaffected if theparent particles decay around thematter-radiation
equality. In this case, the matter density during the radiation-dominated era is mostly contributed by the parent
(cold) particles, rather than by the decay products, and then the density fluctuations of the cold, neutral parent
particles can grow logarithmically. Note that the effect is more pronounced if the parent particles are charged (see
subsection2.6).
– 8 –
1 100
Thermal
0.9 Thermal boson decay
0.8 Non-rel. particle decay
10
0.7
0.6 k=51 Mpc-1
v) 0.5 k) 1 J
g( ∆(
0.4
0.3
0.1
CDM
0.2
Thermal
0.1 Thermal boson decay
Non-rel. particle decay
0 0.01
0 0.5 1 1.5 2 1 10
v/σ k [h Mpc-1]
Fig. 1.— The normalized velocity distributions (left panel) and the dimensionless linear matter
power spectra (right panel) for the standard CDM model and the three WDM models with k =
J
51Mpc−1.
∞
dvv2g(v) = σ2 (8)
Z
0
with the variance (second moment) of the velocity σ2. Note that the first equation is normalized
with respect to the present energy density of WDM, Ω , whereas the second equation relates
WDM
the velocity variance to the comoving Jeans scale at the matter radiation equality k given by
J
Eq.(5). One can then expect that the power spectra for the WDM models with the same Ω
WDM
and k are very similar, as seen in Fig.1. There, we see differences between the WDM models only
J
in the damping tail of the power spectra at k > 10hMpc−1.
2.6. Long-Lived CHAMP and Cut-off scale
Sigurdson & Kamionkowski (2004) formulate the linearized evolution equations for fluctua-
tions in a Long-Lived CHAMP model. They show that the subgalactic-scale matter density fluc-
tuations are damped via a mechanism called “acoustic damping”. The comoving horizon scale at
which CHAMP decays determines the cut-off scale of the matter power spectrum, which is defined
by(Hisano et al. 2006; Kohri & Takahashi 2010)
k = aH| , (9)
Ch t=τCh
where H is the Hubble parameter and τ is the lifetime of CHAMP. Smaller-scale density fluc-
Ch
tuations with k > k enter the horizon before CHAMP decays and can not grow due to the
Ch
acoustic oscillations of CHAMP in the thermal background. On the other hand, larger-scale den-
sity fluctuations with k < k grow logarithmically even after entering the horizon due to the
Ch
gravitational instability as the density fluctuations of CDM. We assume CHAMPs decay in the
– 9 –
radiation dominated era. Then the comoving horizon scale at t = τ is evaluated as,
Ch
τ −1/2 g 1/4
k = 2.2Mpc−1× Ch Ch (10)
Ch
(cid:18) yr (cid:19) 3.363
(cid:16) (cid:17)
where g is the effective number of massless degrees of freedom when CHAMP decays.
Ch
WeneedtoconsiderthreephysicalprocessesfortheCHAMPmodel. First,wedescribetheneu-
tralization of CHAMP. A positively charged particle may become neutral by forming a bound state
withanelectron e. Itsbindingenergyis, however, almost thesameas thehydrogen, E ≃ 13.6eV.
be
Hence, the positively charged particle keeps charged until its decay, since we assume CHAMP de-
cays in the radiation dominated era. A negatively charged particle may become neutral by forming
a bound state with a proton p. Its binding energy E ≃ 25keV is almost m /m ∼ 2000 times
bp 4He p
largerthanE andhenceisexpectedtomakethenegatively chargedparticleneutralatT ∼ 1keV.
be
However, Helium4HeisproducedthroughBBN,withwhichanegatively chargedparticlemayform
a binding state. Its binding energy E ≃ 337keV is almost (Z /Z )2 ×m /m ≃ 16 times
b4He 4He p 4He p
larger than E . It should be noted that even a negatively charged particle bound with a proton is
bp
wrested by 4He through a charge-exchange reaction(Kamimura et al. 2009). Therefore, when the
yield of CHAMP Y (Y ≡ n/s with the number density n and the entropy density s) is smaller
Ch
than the yield of the Helium Y , almost every negatively charged particle forms a binding state
4He
with a helium nuclei, which has one positive elementary charge(Kohri & Takahashi 2010).
Second,thedecayproductsofCHAMPmayleadtoenergyinjectiontothethermalbackground.
TheresultinginjectionenergydensityisconstrainedfromthephotodissociationofBBN(Kawasaki et al.
2001) and CMB y- and µ− parameters(Hu & Silk 1993). However, models with CHAMP with al-
mostthesamemasswithneutraldarkmatterarenotseverelyconstrainedbyBBNnorbyCMB.We
focus on such an “unconstrained” model. Note that the small mass splitting ensures the relatively
long lifetime of CHAMP and the “coldness” of neutral dark matter.
Finally,CHAMPsaretightlycoupledwithbaryonsbeforeitsdecay. Sigurdson & Kamionkowski
(2004) assume θ = θ where θ is the divergence of the fluid velocity. This approximation is
baryon Ch
valid when the Coulomb scattering between baryons and CHAMPs is efficient, i.e., CHAMPs and
baryons are tightly coupled. However, the constraints from the Catalyzed BBN essentially allow
only heavy CHAMP with m & 106GeV for τ & 103sec. It is unclear if the Coulomb scattering
Ch Ch
between baryons and such heavy CHAMPs is efficient. We have calculated the scattering efficiency
and found that the tightly coupled approximation is indeed valid through the epoch of interest
for m . 108GeV. The details of the calculations are found elsewhere(Kamada et al.). In the
Ch
following, we adopt the formulation given by Sigurdson & Kamionkowski (2004) using the tightly
coupled approximation between baryons and CHAMPs.
We modify CAMB(Lewis et al. 2000) to follow the evolution of density fluctuations in the Long-
Lived CHAMP model. The basic equations are given in Sigurdson & Kamionkowski (2004). We
obtain the power spectra for several τ s (see Eq.(10)). We find that the CHAMP matter power
Ch
– 10 –
100
10
k =51 Mpc-1
k) 1 cut
∆(
CDM
0.1
Thermal
Thermal boson decay
Non-rel. particle decay
Long-Lived CHAMP
0.01
1 10
k [h Mpc-1]
Fig. 2.—WeplotthedimensionlesslinearmatterpowerspectruminLong-LivedCHAMPmodel(τ ≃2.5yr).
We compare it with the same dimensionless linear matter power spectra in the WDM models as in Fig.1.
−1
The oscillation around k ∼9hMpc is the imprint of the “acoustic damping”.
spectrum is very similar to the WDM models, as seen in Fig.2, when k is set such that
Ch
k ≡ k ≃ 45k . (11)
cut J Ch
Hereafter, we use k defined in the above as a characteristic parameter of the models we consider.
cut
The corresponding lifetime of CHAMP is τ ≃ 2.5yr in the figure. The imprint of the CHAMP
Ch
“acoustic damping” on the linear matter power spectra is clearly seen. One can naively guess that
structures in the Long-Lived CHAMP model would be similar to those in the WDM models. It is
important to study the non-linear growth of the matter distributions in the Long-Lived CHAMP
model. We use large cosmological N-body simulations to this end.
3. Numerical simulations
Our simulation code is the parallel Tree-Particle Mesh code, GADGET-2(Springel 2005). We
use N = 5123 particles in a comoving volume of L = 10h−1Mpc on a side. The mass of a
simulation particle is 5.67×105h−1M and the gravitational softening length is 1h−1kpc. We
sun
run a friends-of-friends (FoF) group finder(Davis et al. 1985) to locate groups of galaxies. We
also identify substructures (subhalos) in each FoF group using SUB-FIND algorithm developed by
Springel et al. (2001). We do not assign any thermal velocity to simulation particles because it can
lead to formation of spurious objects(Col´ın et al. 2008). We start our simulation from relatively
low redshift z = 19, at which the thermal motion of WDM is redshifted and negligible. It should
be noted that the heavy, neutral dark matter produced by the CHAMP decay is assumed to have
negligible thermal velocities.
In Fig.3, we plot the projected matter distribution in the CDM model (left panel), in the